284 



BELL SYSTEM TECHNICAL JOURNAL 



The complete equation for the horn, equivalent to equation (48), 

 becomes 



p2. 



-LT . 



cosh L-.m- 



O 



+ 



>-?? 



sinh 



(l^ 



C 



TiVP^^sinh^L^r^-^^)] 



Si xJT^ ~ 7^ 



(49) 



Vo = g^ 



cosh L^ T' - 



a 



T 



4^-$ 



^sinh ( 



L^W -^2 



Fi 



iSipi -^ sinh iLyiX^ ~ C^ ) 



^IP^PyJT'-^^ 



These expressions can be derived from Webster's ^ differential equa- 

 tions for an exponential horn. Exponential horns have also been dis- 

 cussed by a number of writers. '" 



B. Tapered Filters Whose Area Increases as the Square of the 



Distance 



One other example of a tapered filter, for which an approximate 

 solution can be obtained, will be considered because of its bearing on 

 the straight or conical horn. Let us assume that the area Si of a 

 typical section of a tapered filter chain is n^E, while that of the section 

 next to it is equal to (n + 1)"'^£, where £ is a small constant. Sub- 



* A. G. Webster, "Acoustic Impedance, and The Theory of Horns and of the 

 Phonograph," Nat. Acad, of Science, Vol. 5, 1919, p. 275. The solution given by 

 Webster for the exponential horn appears to have some typographical errors. 



i»Hanna and Slepian {Trans. A. I. E. E., 43, 1924, p. 393); H. C. Harrison 

 (British Patent No. 213,525, 1925); I. B. Crandall, "Theory of Vibrating Systems 

 and Sound," D. Van Nostrand, 1926, p. 158. 



